Breakup of three-body exotic nuclei

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Edna Carolina Pinilla Beltr´ an

Facult´e des Sciences Universit´e libre de Bruxelles

Thèse de doctorat présentée en vue de l’obtention du grade de Docteur en Sciences Physiques

Novembre 2012

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To my lovely parents, sister, brother and husband

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Bragg, Sir William

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Acknowledgements

Ending this stage of my life would have been possible without the support of many people. First and foremost, I would like to express my sincere gratitude to my advisor Pierre Descouvemont for believing in me, for his patient and dedication along all this research period. His constant encouragement, knowledge, expertise, support and invaluable suggestions made this work successful.

It is a great pleasure to thank the PNTPM members, especially to Daniel Baye for all his contributions, advices, support and great experience. I am particularly grateful for the assistance given by Pierre Capel. I also would like to highlight the kindness of Veerle Hellemans, Simone Baroni, Horacio Olivares, Jeremy Dohet- Eraly and Thomas Druet. I wish to my brother-in-arms Jeremy and Thomas an excellent ending of their dissertation.

I would like to thank to my Jury committee members Nathalie Vaeck, Angela Bonaccorso, Jean-Marc Sparenberg, Claude Semay and Michele Sferrazza, for their time and comments about this thesis.

I am deeply and forever indebted to my parents for their love, support and encouragement throughout my entire life. I am also very grateful to my sister and my brother for making my life happier. Special thanks to my husband and best friend Oscar who supported me in every possible way to complete this work. I also like to thank my friends Nancy, Robin, Germ´an, Ricardo and Gabo for their online encouragement.

This research would not have been possible without the financial support of Pˆoles d’attraction Interuniversitaires (PAI) P6/23 programme and the Institute Interuni- versitaire des Sciences Nucleaires. FNRS/IISN.

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1. E. C. Pinilla and P. Descouvemont, Phys. Lett. B686 (2010) 124-126 2. E. C. Pinilla and P. Descouvemont, Nucl. Phys. A834 (2010) 499c-501c

3. E. C. Pinilla, D. Baye, P. Descouvemont, W. Horiuchi and Y. Suzuki, Nucl. Phys.

A 865 (2011) 43-56.

4. E. C. Pinilla, P. Descouvemont and D. Baye, Phys. Rev. C85(2012) 054610.

5. P. Descouvemont, E. C. Pinilla and D. Baye, Prog. Theor. Phys. Suppl. 196(2012), 1-15.

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Introduction

Nuclei are quantum systems characterized by a mass number A, a charge number Z and a neutron number N. The binding energy B in a nucleus is the energy needed to separate all the nucleons. If just part of the nucleus is taken away, the energy necessary for such process is called separation energy. Stable nuclei are characterized by large binding energies (B/A∼8 MeV) and extremely long lifetimes, longer than the expected lifetime of the solar system (about 4.6 billion of years). Out of the thousand of known nuclear species, about 300 hundred are stable; that is they exist along the valley of stability (black squares in Figure 1.1) [1].

Figure 1.1. Chart of light nuclides up toN = 22 andZ = 10. The driplines are based on identified bound nuclei. Neutron and proton halo nuclei are shown by circles. The yellow squares indicate neutron rich nuclei and the blue ones proton rich nuclei. Figure edited from Ref. [2].

By adding neutrons, a chain of nuclear isotopes is formed. As an isotope moves away from the valley of stability, the separation energy of its(their) last neutron(s) decreases until eventually it reaches zero and bound isotopes do no longer exist [3]. In a plot of the proton number versus the neutron number of nuclei, the boundary where the separation energy of the valence neutron(s) is zero, is called neutron dripline. Similarly, the addition of protons from the valley of stability to a nuclear system will lead to the proton dripline.

Nuclei far from the stability valley are referred to as exotic nuclei. They present very

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interesting features, different from the stable nuclei [4]. Close to the driplines there are exotic systems calledhalo nuclei [5,6] (see Figure 1.1). Such nuclei exhibit an anomalously large radius in comparison with their neighbors, appreciably larger than the tendency r∼1.2A^1/3fm. This fact has been interpreted as a halo phenomenon, in which the nucleus is seen as made of a core and of a low density cloud of valence nucleon(s) surrounding the core and located far from it [7]. Halo nuclei were discovered with the advent of new experimental techniques capable to produce radioactive nuclei with exotic proton and neutron numbers [8, 9].

The small binding energy of the valence nucleon(s) is responsible for the large extension of halo nuclei. For one neutron halo nuclei, this fact can be understood in a very simplified picture of the neutron embedded in the mean field of the core. Then, if we consider as a mean field potential a square well, the wave function of the halo neutron has an exponential decreasing behavior. That wave function is more and more extended as the separation energy of the valence nucleon decreases [10].

A direct consequence of the weakly bound character of halo nuclei is that they have only one or few bound states. Therefore, any excitation of such a weakly bound system tends to be into the continuum. Another criterion to distinguish a halo nucleus is that the valence nucleon(s) must have a very low angular momentum relative to the core, preferablyl= 0, since higher values will provide a confining centrifugal barrier [5]. Indeed, the confining Coulomb barrier makes proton halo nuclei less extended than neutron halo nuclei.

Borromean nuclei are a special kind of halo nuclei [11]. They have a two nucleon halo and can be seen as three-body systems made of a core and two loosely bound nucleons.

In Borromean nuclei no pair core-nucleon or nucleon-nucleon is bound. Their name comes from the Borromean rings or a system of rings that are interlocked in such a way that if any is removed the system will break down. In Figure 1.1 the Borromean nuclei are indicated by two concentric circles. Typical examples, and probably the most studied, are the⁶He and¹¹Li nuclei seen as an α+n+nand ⁹Li+n+nsystems, respectively. They have only one bound state, situated under the three-body breakup threshold. The energy and half-lifeτ_1/2 of this state is 0.973 MeV andτ_1/2∼0.8 s for ⁶He, and, 0.378 MeV and τ_1/2 ∼9 ms for¹¹Li.

In order to describe the structure of halo nuclei, microscopic [12–14] and non-microscopic models [15–18] have been implemented. Microscopic models are based on nucleon-nucleon interactions. Their advantage is that they can take properly the Pauli principle into account. Non-microscopic models are simpler. They visualize the halo nucleus as made of a core plus nucleons. Then, the many-body problem reduces to a few-body problem.

Non-microscopic models are built on nucleus-nucleus or nucleus-nucleon interactions. The Pauli principle in those models is taken approximately by a suitable choice of the nucleus- nucleus or nucleus-nucleon interaction [19–21]. However, those few-body models are easier to interpret and to integrate in reaction models [22–24].

Since halo nuclei are weakly bound, they are easily broken up in the nuclear and Coulomb fields of the target. Hence, breakup reactions have become the most used tool to study such nuclei [25–28]. Coulomb breakup of halo nuclei is a process in which a halo projectile passing by the Coulomb field of a heavy target is excited, with the subsequent breaking up of the projectile into a core plus one or two fragments. The probability distribution of electric dipole excitation to the continuum is called E1 strength distribution.

In most of the Coulomb breakup experiments of halo nuclei an enhancement of the E1 distribution shows up at low excitation energies [25, 29, 30].

Breakup experiments of halo nuclei are typically performed at energies where the eikonal method is suitable to describe the reaction framework. The eikonal method, origi-

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Introduction 3 nally introduced by Glauber [31], makes some simplifying assumptions in the high energy regime, that is, at energies much higher than the Coulomb barrier. Different versions and generalizations have been implemented since Glauber’s publication [32–37]. The breakup cross section in traditional eikonal methods is known to diverge due to the presence of the Coulomb interaction [13]. The Coulomb-corrected eikonal method [33, 34] is proposed to avoid such divergence. This method has been applied, for instance, to study the breakup process of the halo nucleus ¹¹Be in a non-microscopic, two-body description of the ¹¹Be projectile, which is seen as¹⁰Be core plus a loosely bound neutron [33, 34, 37]. Extension to a three-body projectile, the four-body Coulomb-corrected eikonal method, has been started in Ref. [22]. In that reference, the method is applied to the elastic scattering and breakup of the⁶He=α+n+nnucleus on a structureless²⁰⁸Pb target.

An important ingredient in four-body reaction calculations is the three-body projectile wave functions. Various techniques have been applied in three-body problems such as the Faddeev formalism [15], the Gaussian expansion method [38] or the hyperspherical formalism [15, 16, 39]. In the hyperspherical formalism, the three-body Schr¨odinger equation is reduced to a set of coupled differential equations depending on a single coordinate called the hyper-radius. This method is especially well adapted to Borromean nuclei since the three bodies are treated on the same footing.

A reliable description of the breakup process must include an accurate initial, bound state, and final, continuum state, wave functions of the projectile, and an appropriate reaction model. For three-body systems, the treatment of bound states generally relies on variational calculations, which are fairly simple and well known [13, 16, 40]. Besides, the correct asymptotic description of continuum states is expected to play an essential role in the study of the breakup of halo nuclei, due to the large extension of the ground state of those nuclei. For three-body systems, a correct description of continuum states is a chal- lenging task. In Ref. [17], a three-body R-matrix method in the hyperspherical formalism has been introduced with this aim. This treatment is rather difficult and time consuming.

In the literature, simpler approximation methods that discretize the continuum, such as the pseudostate [41, 42] and the complex scaling [43–45] methods are widely used.

In the four-body Coulomb corrected eikonal method of Ref. [22], the wave functions of the projectile are calculated with a three-body model in hyperspherical coordinates. In particular, the three-body R-matrix method of Ref. [17] is used to find the continuum states of ⁶He. In addition to the elastic and breakup cross sections, the E1 strength is computed with those states. Such a calculation predicts an enhancement at low excitation energies of the dipole strength distribution.

The equivalent photon method [46] is often used to describe theoretically the breakup of halo nuclei [41, 43, 44, 47] and to extract, from the experimental breakup cross section, the E1 strength distribution [25]. This method assumes that the halo nucleus is excited by E1 multipolar excitations and that the breakup process is Coulomb dominated i.e. that the nuclear interaction between the projectile and the target plays no role for inducing the breakup. With these assumptions, the breakup cross section is expressed in terms of the E1 strength distribution [13] which only depends on the projectile properties. The four-body Coulomb-corrected eikonal of Ref. [22] is more general since it considers the nuclear and Coulomb interactions, as well their interference, consistently. Additionally, it does not make any restriction about the multipolarity of the excitation and allows to compute the E1 strength distribution and the breakup cross section separately.

The present work relies in extending the work started in Ref. [22]. As a first part, we continue the study of the E1 strength of⁶He. We compute this E1 strength by using the pseudostate and complex scaling methods that discretize the continuum. The goal is

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to compare those distributions with the one that uses the R-matrix. Since the R-matrix method is more precise, we take it as a reference calculation to assess the validity of the approximation methods [48, 49]. The existence or not of an enhancement in the dipole strength is also discussed.

In the second part of the work, we extend the application of the four-body eikonal method to describe the elastic scattering and breakup of the ¹¹Li=⁹Li+n+n nucleus impinging on a ²⁰⁸Pb target at 70 MeV/nucleon [50]. We first discuss the existence of three-body resonances in the prominent partial waves of the continuum. Then, we calculate the breakup cross section and angular distribution and compare them with the experimental data of Ref. [25]. Additionally, the E1 strength is computed by using the pseudostate and the R-matrix methods. These theoretical calculations are compared with the experimental data [25]. The experimental determination of the E1 strength from the breakup cross section is particularly discussed.

As a third part of the work, we discuss the possibility to introduce a microscopic description of the projectile in the eikonal method. This exploratory work is aimed at describing breakup reactions with a more precise projectile wave function. As a first step, we start with a simple α projectile colliding elastically on ⁵⁸Ni and ²⁰⁸Pb targets at 72 MeV/nucleon [51, 52].

The text is organized as follow: Chapter 2 gives some fundamentals about reaction theory. In particular, it introduces, in a single channel context, the R-matrix method to calculate continuum states with the correct asymptotic behavior, and the pseudostate and complex scaling methods that discretize the continuum. Chapter 3 is devoted to the three- body model in hyperspherical coordinates, to compute bound and continuum states as well the dipole strength distributions with the R-matrix and discretization methods. The reaction theoretical framework to be applied to ¹¹Li on ²⁰⁸Pb, the four-body Coulomb- corrected eikonal model, is presented in Chapter 4. In Chapter 5, the electric dipole strength distribution of⁶He is calculated within the three-body model described in Chapter 3 for the R-matrix, pseudostate and complex scaling methods. Chapter 6 presents the analysis of the elastic and breakup reactions of ¹¹Li on ²⁰⁸Pb. Chapter 7 shows some results of the exploratory work to include a microscopic description of the projectile in nuclear reactions. We consider a simple projectile, an “α-particle”, where each nucleon spatial part is a 0s harmonic oscillator shell orbit [53]. Finally, Chapter 8 is devoted to concluding remarks and outlook.

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Chapter 2

Fundamentals of reaction theory

If we wish to investigate the properties of complex microscopic systems such as nuclei, molecules or atoms, we must solve the many body Schr¨odinger equation which involves many degrees of freedom. This is a very difficult task, so we often simplify the problem reducing the bodies or interactions that compose each system.

We can define two non-composite interacting systems like those for which their internal constituents are not taken explicitly into account, but their existence is simulated by effective two-body potentials. We can consider in the description of non-composite objects, internal degrees of freedom as spin, or orbital angular momentum. Composite objects are made of non-composite objects, that in general, are not their most fundamental constituents.

If the interaction is central, we can study the scattering process through a partial wave expansion of the relative projectile-target wave function. This is the subject of Section 2.2, where we study the elastic scattering of two spinless non-composite nuclei.

At high energies, many partial waves must be included. Therefore, approximate methods as the Born approximation (see Refs. [54, 55] and References therein) or the eikonal approximation (see Chapters 4 and 6) have been developed.

If we collide two composite objects, as a result of the collision, different processes may take place in addition to elastic scattering. We will describe them in the context of multichannel theory in Section 2.3. For details of Sections 2.2 and 2.3 we refer the reader to texts on quantum mechanics or scattering theory, for instance Refs. [56–58]. In Section 2.4, we briefly explain the R-matrix method [59, 60], which is devoted to solve the Schr¨odinger equation with the correct asymptotic behavior of the wave function, mainly applied to scattering problems. Approximation methods that extend traditional bound state calculations to continuum states are explained in Section 2.5. For the sake of clarity, Sections 2.4 and 2.5 are explained in a single channel context.

2.1 Elastic scattering

For typical dimensions of the collision, it is a good approximation to consider stationary scattering states to study the scattering process instead of wave packets as it formally must be [61]. If the collision involves non-composite charged particles, let us define the stationary scattering states with ingoing (−) and outgoing (+) boundary conditions as

Ψ^(±)_k (r)−−→_r>a 1 (2π)^3/2

"

e^i(k^·^r+η^ln(kr−^k^·^r))+f_k(Ω)e^±^i(kr⁻^η^{ln 2kr)} r

#

, (2.1)

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where r > a is the typical distance where the detector in the collision is located. The above expression means, that far from the scattering center, the solution of the scattering problem corresponds to a plane wave distorted by the presence of the Coulomb potential, plus a spherical wave, also distorted by the Coulomb potential, and modulated by the scattering amplitudef_k(Ω).

One can obtain the ingoing stationary scattering states from the outgoing ones through the time reversal operation

KΨ⁽⁻⁾_k (r) = Ψ⁽⁺⁾₋_k(r), (2.2) withKthe time-reversal operator.

The elastic cross section can be found from the elastic scattering amplitude as dσ

dΩ =|f_k(Ω)|², (2.3)

with Ω the solid angle of the detector.

2.2 Partial wave expansion

Let us consider two non-composite nuclei interacting via a central potential only. Then, its Hamiltonian is given by

H=

2

X

i=1

p²_i

2m_i +V(|r₂−r₁|), (2.4) where the first term is the kinetic energy and the second is the central interaction between both nuclei.

After removing the center of mass motion, the Hamiltonian in Eq. (2.4) is written in terms of the relative coordinater=r2−r1 as

H =−~²

2µ∇²r+V(r), (2.5)

withµ the reduced mass of the two nuclei and r =|r|. The relative wave function Ψ(r) satisfies the stationary Schr¨odinger equation

HΨ(r) =EΨ(r). (2.6)

Because of the spherical symmetry of the potential, the Hamiltonian commutes with the angular momentum operators ˆL² and ˆL_z. Therefore, it is convenient to express Ψ(r) as the following combination of partial waves

Ψ(r) = 1 (2π)^3/2

X

lm

C_lmΨ^lm(r),

= 1

(2π)^3/2 X

lm

C_lmY_lm(Ω)R^l(r). (2.7)

Here the Y_lm(Ω) are the spherical harmonics, Ω = (θ, ψ) is the solid angle and R^l(r) is chosen as

R^l(r) = χ^l(r)

kr . (2.8)

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2.2. Partial wave expansion 7 After introducing the expansion (2.7) into the stationary Schr¨odinger equation (2.6), we end up with the radial wave equation for each partial wave

−~² 2µ

d² dr² + ~²

2µ

l(l+ 1)

r² +V(r)

χ^l(r) =Eχ^l(r), (2.9) where ^~_2µ²^l(l+1)_r2 is the centrifugal potential.

Until here the presentation is rather general. Let us take the two non-composite nuclei as spinless and charged. Hence, the interaction potential has the form

V(r) =V_N(r) +V_C(r), (2.10)

whereV_N(r) is the nuclear interaction which goes to zero faster than 1/r² when r → ∞ andV_C(r) is the Coulomb potential. If we consider the finite size of one of the nuclei, the Coulomb potential can be approximated as a point nucleus of chargeZ₁e, interacting with a charged sphere of chargeZ2e. This potential is written as

V_C(r) = (_Z

1Z2e² 2R_C

3−_R^r²2 C

forr < r_C,

Z1Z2e²

r forr ≥r_C, (2.11)

withR_C =r_CA^1/3₂ , A₂ the mass number of the target and r_C a parameter which typical value is 1.2 fm. Here and in the following, we will takee² ≡ _4πǫ^e²₀.

If we consider the two nuclei as two point charges, the radial equation (2.9) becomes d²

dr² −l(l+ 1) r² −2kη

r − 2µ

~²V_N(r) +k²

χ^l(r) = 0, (2.12) where η = ^Z¹^Z_~2²k^e²^µ is the Sommerfeld parameter which measures the strength of the Coulomb interaction andE = ^~_2µ²^k².

In order to find solutions of Eq. (2.12), it is useful to divide the configuration space into two regions: i) The internal region with r ≤ a, where the nuclear and Coulomb interaction dominate. ii) The external region with r > a, such that a is large enough to consider that the nuclear potential vanishes. This division of the configuration space allows the implementation of boundary methods to solve the Schr¨odinger equation such as the R-matrix method (see Section 2.4).

2.2.1 Partial wave expansion of scattering states

Solutions of the Schr¨odinger equation Ψ(r) with energies E >0 are known as scattering states. A partial wave Ψ^lm_k (r) of a scattering state is not square integrable but its norm is fixed by the property

hΨ^l_k^′^′^m^′|Ψ^lm_k i=δ(k−k^′)δ_ll^′δ_mm^′. (2.13) Now, let us come back to the radial equation (2.12) for scattering states. In this case, the radial functionχ^l_k(r) must satisfy the boundary conditions

χ^l_k(0) = 0, (2.14a)

χ^l_k(r > a) = i 2e⁻^iδ^lh

H_l⁽⁻⁾(η, kr)−U_lH_l⁽⁺⁾(η, kr)i

, (2.14b)

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withU_l =e^i2δ^l a partial wave scattering matrix element andδ_l called the phase shift. The Coulomb Hankel functions of real argumentH_l^(±)(η, kr) are known as incoming (−) and outgoing (+) Coulomb waves because of its asymptotic behavior

H_l^(±)(η, r > a)→e^±i(kr−^lπ₂−ηln 2kr+σl), (2.15) which represent outgoing or ingoing waves distorted by the Coulomb potential. Theσ_l = σ₀+P^l

s=0

tan⁻¹(^η_s) is called Coulomb phase whereσ₀= argΓ(1 +iη).

The Coulomb Hankel functions can be expressed in terms of the Coulomb functions F_l(η, kr) and G_l(η, kr) as

H_l^(±)(η, kr) =G_l(η, kr)±iF_l(η, kr). (2.16) Therefore, the radial wave function (2.12) becomes

χ^l_k(r > a) =G_l(η, kr) sinδ_l+F_l(η, kr) cosδ_l. (2.17) Let us come back to Eq. (2.12) where we can recognize three limiting cases: i) We switch off the nuclear potential, ii) we switch off the Coulomb potential and iii) we switch off both Coulomb and nuclear potentials. For the first case, we have just the Coulomb interaction and therefore, Eq. (2.12) is reduced to the Coulomb equation [62] where the Coulomb functions F_l(η, kr) and G_l(η, kr) are their regular and irregular solutions respectively. In the second case, Eq. (2.14b) is written in terms of the Hankel functions, H_l^(±)(0, kr), which can be derived from

F_l(0, kr) = (πkr/2)^1/2J_l+1 2(kr), G_l(0, kr) =−(πkr/2)^1/2Y_l+¹

2(kr), (2.18)

withJ_l+¹

2(kr) andY_l+¹

2(kr) the Bessel functions of the first and second kind respectively.

The last case, corresponds to have no potential. We can show that changingr→ kr, the Eq. (2.12) becomes the Ricatti-Bessel equation [62], where the regular and irregular solutions are the Ricatti Bessel ˆj_l =krj_l(kr) and Ricatti-Neumann ˆn_l =krn_l(kr) functions.

The spherical Bessel functionsj_l(kr) and n_l(kr), can be expressed in terms of the Bessel functionsJ_l+¹

2(kr) and Y_l+¹

2(kr) as j_l(kr) = π

2kr 1/2

J_l+1

2(kr), n_l(kr) = π 2kr

1/2

Y_l+1

2(kr). (2.19) 2.2.2 Phase shifts

In order to clarify the meaning of the phase shiftδ_l, it is convenient to consider the radial function R^l_k(r) in the external region for the case ii). Then, if we use the asymptotic behavior of the Coulomb functions withη= 0

F_l(0, r > a) = sin

kr−lπ 2

, G_l(0, r > a) = cos

kr−lπ 2

, (2.20)

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2.2. Partial wave expansion 9 in Eq. (2.17), the radial wave function (2.8) becomes

R^l_k(r > a) = π 2kr

1/2h J_l+1

2(kr) cosδ_l−Y_l+1

2(kr) sinδ_li ,

= sin kr−^lπ₂ +δ_l

kr . (2.21)

On the other hand, if we also switch off the nuclear potential, Eq. (2.21) must be valid in the whole range of r. In particular, it must be valid at r= 0. Near the origin, the Bessel function of the first and second kind behaves as

J_l+1

2(kr →0)≃ 2kr

π 1/2

(kr)^l (2l+ 1)!!, Y_l+1

2(kr →0)≃ 2kr

π

1/2(2l−1)!!

(kr)^l+1 . (2.22)

Therefore, considering those limits in Eq. (2.21), we can see that in absence of potential, the wave function diverges except when the phase shiftδ_lis zero. Hence, the radial function reduces to

R^l_k(r) =sin kr− ^lπ₂

kr . (2.23)

By comparing Eq. (2.21) and Eq. (2.23), we observe that the effect of the short range potential is a shift of the phase in the radial wave function. It turns out thatδ_lis positive (negative) for attractive (repulsive) short range potentials.

The phase shifts are the physical quantity of interest. They are obtained from the matching of the internal and external radial wave functions and their derivatives ina. In other words, when the internal wave function is found, either by numerical integration or by means of a variational expansion, it is matched in awith the external radial function (2.14b) to find the phase shift for eachl.

2.2.3 Partial wave expansion of the elastic cross section

In order to find the partial wave decomposition of the elastic scattering amplitude for non-composite charged nuclei, let us rewrite it as

f_k(Ω) =f_N(Ω) +f_C(Ω), (2.24)

where f_N(Ω) is a correction to the Coulomb scattering arising from the short range potential andfC(Ω) is the Coulomb scattering amplitude which is given by

fC(θ) =− η

2ksin²θ/2e⁻^iη^ln(sin²^θ/2)e^2iσ⁰, (2.25) for an incident projectile nucleus withk=kz.

By using Eq. (2.24), we can rewrite Eq. (2.1) for the outgoing boundary condition as Ψ⁽⁺⁾_k (r)−−→

r>a Ψ^C(+)_k (r) + 1 (2π)^3/2

"

f_N(Ω)e^i(kr−η^{ln 2kr)} r

#

, (2.26)

with Ψ^C(+)_k (r) the asymptotic form that must have the solution of the Schr¨odinger equation when just the Coulomb potential is present [54].

Asf_C(Ω) is known, our problem is reduced to find the partial wave decomposition of f_N(Ω). This decomposition reads [54]

f_N(θ) =X

l

(2l+ 1)P_l(cosθ)

2ik e^2iσ^l(U_l−1). (2.27)

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2.2.4 Resonances

To explain intuitively the concept of resonance, let us come back to Eq. (2.9) and define the effective potential as

V_l^eff(r) =V_N(r) +V_C(r) + ~² 2µ

l(l+ 1)

r² . (2.28)

This potential has a finite well plus a finite barrier and therefore it is possible that a particle of reduced mass µ could be trapped inside the well, but not forever. The state of the particle is sometimes called quasibound state, coming from the fact, it will be a genuine bound state if the barrier were infinitely high. As the particle cannot be trapped forever because of the finite barrier it has a finite lifetimeτ.

One feature of a resonance state is that the corresponding phase shift rises rapidly to π/2 as the incident energyE_Incident rises and it is ∼π/2 whenE_Incident ∼E_R. The cross section for a given partial wave as a function of energy, called excitation function, also exhibits a peak often related with the existence of a resonance.

Let us see how is the behavior of the total cross section, for a given partial wave, when the energy is close to the energy of the resonance, and just the nuclear interaction acts.

In this case, the total cross section reads [55]

σ= 4π k²

X

l

(2l+ 1) sin²δ_l(E). (2.29)

Here, we bring out explicitly the dependence of the phase shift on energy. Then, a partial wave cross section can be written as

σ_l= 4π

k²(2l+ 1) 1

1 + cot²δ_l. (2.30)

Expanding cotδ_l(E) up to first order around the resonance energy E_R, we have cotδ_l(E)≃ ∂cotδ_l(E)

∂E E=ER

(E−E_R). (2.31)

If we define Γ/2 =

∂cotδl(E)

∂E

E=E_R

₋₁

and introduce the relation (2.31) into Eq. (2.30) we end up with the Breit-Wigner form for an isolated resonance

σ_l ≃ π(2l+ 1) k²

Γ²

Γ²/4 + (E−ER)². (2.32) From it, Γ, the resonance width, can be seen as the full width at the half-maximum of the resonance peak and it is related with the lifetime of the resonance state as Γ∼~/τ. 2.2.5 Bound states

A bound state Ψ^lm_n (r) satisfying (2.6) is such that for a givenlhas a energyE_nl<0. The bound states fulfill the orthonormality property

hΨ^l_n^′^m^′ ^′|Ψ^lm_n i=δ_nn^′δ_ll^′δ_mm^′. (2.33) Here we have introduced the indexnto indicate the discrete character of the bound states spectrum.

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2.3. Multi-channel theory 11 Scattering states are orthogonal to the bound states

hΨ^l_n^′^m^′ ^′|Ψ^lm_k i= 0, (2.34) and together satisfy the closure relation

X

nlm

|Ψ^lm_n ihΨ^lm_n |+X

lm

Z _∞

0

dk|Ψ^lm_k ihΨ^lm_k |= 1. (2.35) For the specific case of Eq. (2.12) the radial function fulfills the boundary conditions [63]

χ^l_n(0) = 0, (2.36a)

χ^l_n(r > a) =C_lW_−η_I_,l+¹

2(−2kIr)−−−→

r≫a C_le⁻^kÎ^r⁻^ηÎ^ln(2kÎ^r), (2.36b) with the notations

k=ik_I =i

r2µ|E_nl|

~² and η=−iη_I =−iZ₁Z₂e²µ

~²k_I . (2.37) The functionW_−η_I_,l+¹

2(−2k_Ir) is the Whittaker function given by [62]

W₋_η_I_,l+1

2(−2k_Ir) =eîπηÎ^/2e⁻î(πl/2+σ^l^(η)H_l⁽⁺⁾(−iη_I, iK_Ir), (2.38) where the coefficient C_l is known as the Asymptotic Normalization Coefficient (ANC) which describes the strength of the exponential tail. TheH_l⁽⁺⁾(−iη_I, iK_Ir) and σ_l(−iη_I) are the analytical continuation to the complex plane of the Coulomb Hankel and Coulomb phase functions for imaginary argument respectively [62].

In the case of neutral nuclei, the radial function (2.36b) has the familiar exponential decreasing behavior

χ^l_n(r)−−−→

r≫a C_le⁻^K^I^r. (2.39)

2.3 Multi-channel theory

The theory developed so far where neither the target nor the projectile have internal structure must be extended to the more realistic situation where excitation, mass rearrangement, between other processes can happen. For instance ifAandB are two colliding composite objects, after the collision, we can have different processes as

A+B →A+B, (2.40a)

A+B →A+B^∗, (2.40b)

(C+D) +B →C+ (D+B); A= (C+D), (2.40c) (E+D+F) +B →E+D+F +B; A= (E+D+F). (2.40d) The sets in parentheses correspond to bound systems. The first equation above refers to elastic scattering, the second to excitation of one of the systems, the third to mass rearrangement and the last one is called breakup. Strictly speaking processes (2.40c) and (2.40d) are called reactions but it is common to denominate (2.40a) and (2.40b) as reactions as well.

When the possible mass rearrangements of the participants objects are specified, the quantum numbers of their relative motion, and their intrinsic quantum numbers, one

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speaks about a reaction channel. A reaction proceeds from the entrance channel to various exit channels. For instance, in the cases (2.40), the mass partitions from the left belong to the entrance channel, and the mass partitions from the right to the possible different exit channels. If the total collision energy is high enough for an exit channel to be possible, we say that the channel is open. Otherwise, we say it is closed.

In the previous example, the systemsAandBbelonging to the entrance channel satisfy HÂΨÂ(ξa) =ǫÂΨÂ(ξa),

H^BΨ^B(ξ_b) =ǫ^BΨ^B(ξ_b), (2.41) where Ψ^A(ξA) and Ψ^B(ξB) are the internal wave functions of the systemsA and B with intrinsic coordinates ξ_A and ξ_B respectively. The subscript aand b represents the set of quantum numbers associated with each system including its parity. In a compact notation we define the intrinsic wave function of the projectile and the target

Ψ_c^′(ξ) = Ψ^A(ξ_a)Ψ^B(ξ_b), (2.42) which fulfills the Schr¨odinger equation

h_c^′Ψ_c^′(ξ) =ǫ_c^′Ψ_c^′(ξ), (2.43) with h_c^′ = H^A+H^B, ǫ_c^′ = ǫ^A+ǫ^B and ξ representing the internal coordinates of the projectile and the target. The indexc^′ will be explained hereafter.

By considering the relative coordinate from the center of mass of the projectileA and the target B as r_α = r_A−r_B, and after removing the center of mass motion, the total Hamiltonian can be written as

H_c^′ =− ~²

2µ_α ▽²α+h_c^′+V_α(r_α, ξ). (2.44) The first term is the relative kinetic energy with reduced mass µ_α = _m^m^A^m^B

A+mB, where m_A and mB are the projectile and target masses. The last term is the interaction between both systems defined as

V_α(r_α, ξ) = X

i∈A,j∈B

V_ij(r_α, ξ_a, ξ_b). (2.45) The c^′ labels the entrance channel. For instance if both projectile and target have spin, c^′ is the set c^′={A, B, a, b, l_α, S_α, J}, withA, B indicating that we have a mass partition with two systemsAand B,l_α is the relative angular momentum, S_α the total spin of the channel andJ is the total angular momentum of the system which is a conserved quantity.

Let us suppose we may have two composite objects in an exit channelc. In our example, this situation corresponds to the cases (2.40a), (2.40b) and (2.40c). Then, after removing the center of mass motion, the Schr¨odinger equation of any of those cases is given by

H_cΦ_c^′ =E_TΦ_c^′, (2.46)

where the subscript in the wave function indicates that the collision initiated in channel c^′. The Hamiltonian H_c has the same form of Eq. (2.44) replacing α by β and c^′ by c.

The total energy of the system is given by E_T =ǫ_c+

~²k_β²

2µ_β . (2.47)

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2.4. The R-matrix method 13 2.3.1 Coupled-channel equations

Let us expand the wave function Φ_c^′, solution of the Sch¨odinger equation (2.46), in terms of the basis of intrinsic states

Φ_c^′(r_c, ξ) =X

c

ψ_c(c^′₎(r_c)Ψ_c(ξ). (2.48) Here and in the following we substitute the indexβ by c. If we introduce this expansion into the Schr¨odinger equation (2.46) and project onto the intrinsic state, we have

X

c

Z

dξΨ^†_c_˜(ξ)

− ~²

2µ_c ▽²c +h_c+V_c(rc, ξ)−E_T

Ψ_c(ξ)ψ_c(c^′₎(rc) = 0. (2.49) Note that the integral over ξ stands for integration over the internal coordinates of the projectile and the target. Whencand ˜crelate the same mass partition the internal states (2.42) fulfill

Z

dξΨ^†_˜_c(ξ)Ψ_c(ξ) =δ_c,˜_c (2.50) and Eq. (2.49) reduces to

− ~²

2µ_c ▽²c +ǫ_c−E_T

ψ_c(c^′₎(rc) +X

˜ c

V_c˜_c(rc)ψ_c(c_˜ ^′₎(rc) = 0, (2.51) with the potential

V_˜_cc(r_c) = Z

dξ Ψ^†_˜_c(ξ)V_c(r_c, ξ)Ψ_c(ξ). (2.52) 2.3.2 Asymptotic behavior of the multi-channel radial wave functions When the relative wave function ψ_c(c^′₎(rc) is expanded in partial waves, its radial part must satisfy the asymptotic behavior

χ_c(c^′₎(r_c)−−−→_r

c>a Ch H_l⁽⁻⁾

c′ (η_c, k_cr_c)δ_cc^′−U_cc^′H_l⁽⁺⁾_c (η_c, k_cr_c)i

. (2.53)

HereU_cc^′ is a matrix element of the scattering matrixU whose dimension is given by the number of open channels at energyE_T. As the nuclear and Coulomb interactions do not mix the parity we calculate U for each J and π. The normalization constant C plays no role in calculations of the scattering matrix. Note that as defined in Section 2.2 the distanceais large enough to consider that the nuclear potential vanishes.

2.4 The R-matrix method

The R-matrix theory was developed into two different directions: i) The calculable R- matrix which is an efficient technique to solve the Schr¨odinger equation at both positive and negative energies. Indeed, this method is a powerful way to calculate scattering states with the correct asymptotic behavior, specially for three interacting bodies (see Chapters 3, 5 and 6). ii) The phenomenological R-matrix, essentially used in nuclear astrophysics, allows to accurately parametrize low energy cross sections with a small number of parameters. In this text, we will concentrate on the calculable R-matrix and we will refer to it simply as R-matrix method. An extensive review of the two variants can be found in Refs.

[59, 60].

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2.4.1 Calculation of the R-matrix

Let us introduce the R-matrix method in the single channel context and then let us return to Eq. (2.12) of two charged interacting nuclei. This equation can can be rewritten as

(H_l−E)χ_l(r) = 0. (2.54)

Here the radial HamiltonianH_l is given by

H_l=T_l+V(r), (2.55)

whereT_l=−^~_2µ²_dr^d²² +^~_2µ²^l(l+1)_r2 is the kinetic energy andV(r) is the sum of the nuclear and the Coulomb potential.

In the internal region,r ≤a, the radial wave function χ^int_l (r) is expanded in terms of a finite basis of square integrable functions, not necessarily orthogonal,

χ^int_l (r) =

N

X

j=1

cjϕj(r). (2.56)

In order to satisfy the boundary condition χ^int_l (0) = 0, the basis functions must vanish at the origin. In the external region, r > a, the radial function χ^ext_l (r) is taken as the asymptotic wave function (2.36b) for bound states or (2.14b) for scattering states.

The R-matrix for a given energy is defined as R_l(E, B) = χ_l(a)

aχ^′_l(a)−Bχ_l(a), (2.57) where B is a dimensionless boundary parameter. Here and in the following the primes denote derivatives with respect to r. It can be shown [60] that any value of B leads to the same scattering matrix. Therefore, we will set this value as zero for scattering states.

For bound states, we refer the reader to Ref. [60]. If B is zero, the R-matrix is nothing but the inverse of the logarithmic derivative of the radial wave function evaluated at the channel radius.

Over the internal region, the operatorH_l is not Hermitian. Hence, the Bloch operator L= ~²

2µδ(r−a) d

dr (2.58)

is introduced, to makeH_l+L Hermitian over the internal region.

In order to find the R-matrix expressed in terms of the basis functions of the expansion (2.56), the Schr¨odinger equation in the internal region is approximated by the inhomogeneous Bloch-Schr¨odinger equation

(H_l+L−E)χ^int_l (r) =Lχ^ext_l (r). (2.59) Here the approximation relies in writing in the right part of Eq. (2.59) the asymptotic form (2.14b). The above equation is equivalent to have

(H_l−E)χ^int_l (r) = 0,

χ^int_l ^′(a) =χ^ext_l ^′(a). (2.60)

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2.4. The R-matrix method 15 This equation shows, that the Bloch operator not only makesH_l+LHermitian but enforce the smoothness of the radial wave function. Hence, to adequately solve the Schr¨odinger equation, in addition to the last equation, we have to impose

χ^int_l (a) =χ^ext_l (a). (2.61)

On the other hand, inserting expansion (2.56) in (2.59) and projecting the result on the basisϕ_i(r) we have

N

X

j=1

C_ij(E)c_j = ~²

2µϕ_i(a)χ^ext_l ^′(a), (2.62) with the matrix elementsC_ij defined as

Cij(E) = Z a

o

dr ϕi(r)(T+V_l+L−E)ϕj(r). (2.63) By solving Eq. (2.62), the coefficients cj read

cj = ~²

2µχ^ext_l ^′(a)

N

X

i=1

(C⁻¹)jiϕi(a). (2.64) Inserting this coefficients into the expansion (2.56) and using the definition of the R-matrix (2.57), we find

R_l(E) = ~² 2µa

N

X

i,j=1

ϕ_i(a)(C⁻¹)_jiϕ_j(a). (2.65) Here, for single channel scattering the R-matrix corresponds to one element. Extension to multichannel scattering can be done following the same procedure.

From the expression for the coefficients (2.64) and the definition (2.57), applied to the external wave function, we find the internal wave function

χ^int_l (r) = ~²

2µaR_l(E)χ^ext_l (a)

N

X

j=1

ϕ_j(r)

N

X

i=1

(C⁻¹)_jiϕ_i(a), (2.66) that is nothing but a finite-basis approximation of the internal region, connected at the channel radiusacontinuously and smoothly with the external wave function, whose asymptotic behavior is known.

2.4.2 Scattering matrix

Once we know the R-matrix, it is possible to find the scattering matrix appearing in the asymptotic scattering wave function. This is done by using the definition (2.57) and the expression (2.14b) for the external radial wave function. We find

U_l= H_l⁽⁻⁾(η, ka)−akR_l(E)H⁽⁻⁾

′

l (η, ka)

H_l⁽⁺⁾(η, ka)−akR_l(E)H_l⁽⁺⁾^′(η, ka). (2.67)

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2.5 Continuum discretization methods

2.5.1 Pseudostate method

The pseudostate method [64–66] consists in discretizing the continuum. This method is an extension of variational calculations to positive energies. In other words, the discretization is made by expanding the continuum wave function in terms of a basis ofL² functions.

Finding three-body scattering wave functions with the correct asymptotic behaviour is a difficult task (see Chapters 3, 5 and 6). Therefore, the pseudostate method has been used to describe the continuum states of the Borromean nuclei ⁶He and ¹¹Li in a three-body model (see for instance Ref. [41]).

In order to explain the pseudostate method, let us discuss the simple case of two particles interacting through a potential V(r). If we wish to know the solution of the Schr¨odinger equation

HΨ(r) =EΨ(r), (2.68)

at positive energiesE, an approximate solution ˜Ψn(r) in the pseudostate method is found by means of the expansion

Ψ˜_n(r) =

N

X

i=1

c_iϕ_i(r). (2.69)

Here {ϕi} is an orthonormal basis ofL² functions, although a generalization of this presentation is also possible for a non orthonormal basis. This basis, could be for example Gaussian or Lagrange-Laguerre functions (see appendix A.). The wave functions ˜Ψ_n(r) are such that they minimize the quantity

J =hΨ˜n|H−ǫ|Ψ˜ni. (2.70) The coefficientsci are the variational parameters, so that

∂J

∂ci = 0, i= 1,2, ..., N. (2.71)

This condition leads to the systems of equations

N

X

j=1

hϕ_i|H−ǫ|ϕ_jic_j = 0, (2.72) that is nothing but an eigenvalue problem from which we can get the coefficientsc_jfor each eigenvalueǫ_n. Therefore, from the diagonalization of H we have a set ofN_B bound states {Ψ˜^B_n}i=1,2,..NB with negative eigenvalues and a set of pseudostates {Ψ˜^PS_n } with positive eigenvalues.

The pseudostate method can be used in different situations. For instance, it is em- ployed to find transition probabilities to the continuum (see Chapters 3, 5 and 6) or in the Continuum Discretized Coupled Channel (CDCC) framework [67]. In the latter, the pseudostate method is implemented with the aim of reducing an infinite number of coupled differential equations to a discrete set. In this context, let us understand the origin of this approximation, considering as an example, a composite projectile made of two particles interacting with a non composite target. In this presentation, we ignore the spin, isospin and the relative angular momentum. If we take r as the relative coordinate between the two particles in the projectile, andR the relative coordinate between the center of mass of the projectile and the target. The Schr¨odinger equation is written as

[T_R+H_P(r) +V_PT(R,r)] Φ(R,r) =E_TΦ(R,r), (2.73)